Annular ring and non-pneumatic tire

ABSTRACT

The present invention provides an annular beam of monolithic construction of one homogeneous material and a related efficient, low-cost non-pneumatic tire. Specific geometric design, combined with nonlinear elastomer physical properties, enable the suppression of all reinforcing belts, continuous fibers, or other strengthening layers in the annular beam. The annular beam consists of at least two bands that are continuous in the circumferential direction and connected by a web geometry. The non-pneumatic tire consists of the annular beam, a ground contacting portion, a central wheel, and a plurality of web spokes that connect the wheel and beam. When the tire is loaded to a design load against a flat surface over a design contact length, a contact area of essentially uniform pressure is produced, while the load is transmitted from the beam to the hub via tension in the web spokes. The tire can be economically manufactured.

PRIORITY CLAIM

This is a Continuation of application Ser. No. 16/208,916 filed Dec. 4,2018, which is a Continuation of application Ser. No. 15/677,391 filedAug. 15, 2017, which is a Continuation of U.S. patent application Ser.No. 14/304,217 filed Jun. 13, 2014, which claims the benefit of thebenefit of U.S. Provisional Patent No. 61/835,549 filed Jun. 15, 2013.The disclosure of the prior applications is hereby incorporated byreference herein in their entirety. Furthermore, where a definition oruse of a term in a reference, which is incorporated by reference herein,is inconsistent or contrary to the definition of that term providedherein, the definition of that term provided herein applies and thedefinition of that term in the reference does not apply.

TECHNICAL FIELD OF THE INVENTION

The present invention is in the field of non-pneumatic tires and relatedtechnologies. The application scope includes low-cost, efficient designsfor tires. In manufacturing, the field includes high-pressurethermoplastic injection, reaction injection, and cast molding. Thematerial science field involves thermoplastic and thermoset elastomershaving specific nonlinear mechanical properties.

BACKGROUND OF THE INVENTION

Pneumatic tires offer high load capacity per unit mass, along with alarge contact area and relatively low vertical stiffness. High contactarea results in the ability to both efficiently generate high tangentialforces and obtain excellent wear characteristics. However, pneumatictires are also prone to flats. Well known in the patent literature,non-pneumatic tires offer flat-free operation, yet generally containsome compromise.

Higher cost is often associated with non-pneumatic tires when complexassemblages of composite materials are used. Related to this, productionmethodologies can be laborious. For example, U.S. Pat. No. 7,201,194discloses a non-pneumatic tire containing an annular band which iscomprised of at least two inextensible membrane-like reinforcing layers,which are separated by an elastomeric layer. This annular band is thenaffixed to a central wheel via flexible spokes in a web design. Thiscomposite composition suggests a complex manufacturing process. Highproduction costs could be involved.

Conversely, U.S. Pat. No. 6,615,885 discloses a non-pneumatic tire thatcan be fabricated without composite reinforcement. The design consistsof a rim connected to a hub via curved spokes. The spokes aresufficiently rigid such that loads are transmitted via bending. Such astructure works acceptably well for very small tires and low loads;however, one skilled in the art of non-pneumatic structures can showthat this technological approach would result in high tire mass forapplications supporting higher loads in larger-scale applications.

U.S. Pat. No. 7,013,939 discloses a non-pneumatic tire consisting of asimple elastomeric band, a hub, and connecting spokes which actprimarily in tension. As with U.S. Pat. No. 6,615,885, this solutionworks for very small tires and low loads. However, at higher loads withlarger tire dimensions, one skilled in the art of non-pneumatic tiredesign can show that the contact area characteristics become stronglydegraded. This would result in a loss of performance.

US Patent Application US2012/0234444 A1 discloses a non-pneumatic tirewith an annular reinforcing web, made of a homogeneous material.However, the disclosed structure supports load via compression. Thegenerally radial spokes are thick and designed to act as columns undernormal operation. Thus, the distance between the outer tire diameter andthe rigid wheel diameter must be relatively small, in order to resistbuckling under high loads. Therefore, the absorbed energy potential—aprinciple virtue of the pneumatic tire—may be limited in this tiredesign.

Finally, U.S. Pat. No. 8,517,068 B2 discloses a resilient wheel thatoperates similarly to the invention of U.S. Pat. No. 7,201,194, exceptthat the circumferential membranes are connected by discrete cylindricalsupport elements. Here, the advantages could include lower weight andthe ability to overcome temperature limitations associated withelastomers. However, the assemblage of these disparate elements could belaborious and expensive.

The present invention breaks these compromises by disclosing anon-pneumatic tire that, in a preferred embodiment, can be constructedfrom a single homogeneous material, is light-weight, capable of largedeflection, and obtains a large, constant pressure contact area.Structure geometries and non-linear material properties are disclosedthat accomplish the same function as more complicated designs of priorart. In particular, prior art often employs reinforcements that behavelike inextensible membranes, as well as sandwich compositeconstructions. Conversely, the present design is elastomeric, has nomembrane-like behavior, and contains no reinforcement. While theinvention has elegant simplicity, the principles of operation are notreadily apparent to one of ordinary skill in the art of tire design.

While having potentially complex features, the geometries disclosed inthe present invention are generally suitable for realization in thethermoplastic injection, cast, or reaction injection molding process.These practical attributes result in lower cost when produced in largevolume, yet do not come at the expense of the aforementioned primaryperformance attributes of a pneumatic tire.

Second stage processes can be applied to this invention. For example, anon-pneumatic tire with a thermoplastic material can be designed, perprocedures disclosed in this application. Then, a tread material andtread pattern can be affixed to a radially exterior extent of theannular beam. This can be accomplished via retreading procedures, by a2^(nd) stage injection process, or by co-injection.

SUMMARY OF THE INVENTION

A full and enabling disclosure of the present subject matter, includingthe best mode thereof, directed to one of ordinary skill in the art, isset forth in the specification, which makes reference to the appendedfigures.

FIG. 1 is an isometric view of an annular beam of the invention, alongwith the coordinate system defined by the invention.

FIG. 2 is an isometric view of a non-pneumatic tire of the invention.

FIG. 3 is an isometric view of an exemplary embodiment of anon-pneumatic tire of invention.

FIG. 4 is an isometric view of a second exemplary embodiment of anon-pneumatic tire of the invention.

FIG. 5 is an x-z plane equatorial cross-section view of an alternativearrangement of the web connecting two continuous bands in the annularbeam.

FIG. 6 is an equatorial view of a third arrangement of the webconnecting two continuous bands in the annular beam.

FIG. 7 is an equatorial view of a fourth arrangement of the webconnecting two continuous bands in the annular beam.

FIG. 8 is an x-z plane equatorial view of a radial arrangement of theweb spokes that connecting the annular beam to the central hub.

FIG. 9 is an equatorial view of a second arrangement of the web spokes.

FIG. 10 is an equatorial view of a third arrangement of the spokes.

FIG. 11 is an equatorial view of a fourth arrangement of the spokes.

FIG. 12 is an equatorial view of the annular beam, in contact with twoparallel flat contact surfaces.

FIG. 13 is an equatorial view of the tire, in contact with a flatsurface.

FIG. 14 is a Finite Element Model (FEM) of a straight beam with twocontinuous bands connected by webbing geometry subjected to simpleshear.

FIG. 15 is a FEM of a simply supported straight beam with two continuousbands connected by webbing geometry subjected to a uniform distributedload.

FIG. 16 gives a simple example geometry definition of the beam used inthe FEM models of FIG. 14 and FIG. 15 .

FIG. 17 shows FEM results of models from FIG. 14 and FIG. 15 , whenusing geometry definition given in FIG. 16 .

FIG. 18 is a 2D FEM of a tire in accordance with the invention,subjected to a vertical load of 1100 N.

FIG. 19 is closer view of the contact region of the model from FIG. 18 ,showing contact patch force distribution and Von Mises stress.

FIG. 20 is a plot of tensile stress vs. tensile strain for a preferredhomogeneous elastomeric material in accordance with the invention.

FIG. 21 is an isometric view of a third exemplary embodiment of a tireaccording to the invention.

FIG. 22 is an isometric view of a fourth exemplary embodiment of anon-pneumatic tire of the invention.

FIG. 23 is a process flow map of a design procedure for a tire inaccordance with the invention.

DETAILED DESCRIPTION OF THE INVENTION

Reference will now be made in detail to embodiments of the invention,one or more examples of which are illustrated in the Figures. Eachexample is provided by way of explanation of the invention, and notmeant as a limitation of the invention. For example, featuresillustrated or described as part of one embodiment can be used withanother embodiment to yield still a third embodiment.

Definitions

The following terms are defined as follows for this disclosure, withmaterial properties referring to those at ambient temperature, unlessotherwise noted:

“Wheel” or “Hub” refers to any structure for supporting the tire andcapable of attachment to a vehicle axis, and such terms areinterchangeable herein.

“Modulus” means Young's tensile modulus of elasticity measured per ISO527-1/-2. “Young's Modulus,” “tensile modulus,” and “modulus” are usedinterchangeably herein.

“Secant Modulus” is the tensile stress divided by the tensile strain forany given point on the tensile stress vs. tensile strain curve measuredper ISO 527-1/-2.

“Shear Modulus” means the shear modulus of elasticity, calculated byEquation 10, below.

“Tensile Strain at Break” means the tensile strain corresponding to thepoint of rupture as measured by ISO 527-1/-2.

“Flex Fatigue” is the flexural stress fatigue limit, measured by ASTMD671.

“Design Load” of a tire is the usual and expected operating load of thetire.

“Design contact length” is the contact length over a substantially flatsurface when loaded to design load.

The coordinate system defined by the invention is shown in FIG. 1 :

-   -   The circumferential direction is the X direction.    -   The axial direction is the Y direction, coincident with the        rotation axis 140.    -   The radial direction is the Z direction.

The tire equatorial plane is the plane defined by the x-z axes.

DETAILED DESCRIPTION

A non-pneumatic tire employing an annular beam of homogeneous materialis represented herein by structural two dimensional Finite ElementModels (FEM). This section first provides an overall description of theinvention; then, a preferred design practice for developing a tirehaving desired performance characteristics will be disclosed. Thisprocedure employs FEM.

FIG. 1 shows an isometric view of an annular beam 100. The beam definesa radial direction (Z), a circumferential direction (X) and an axialdirection (Y), which is coincident with the axis of rotation 140. Thebeam is of monolithic construction, composed of one homogeneousmaterial. The beam consists of at least a first circumferentiallycontinuous band 120, lying on a radially inward extent of a webstructure 110. At least a second continuous band 130 lies on a radiallyoutward extent of the web structure. In a preferred embodiment, the beamis constructed of an elastomer of specific non-linear mechanicalmaterial properties, and having a modulus of at least 90 MPa, andpreferably between 150 to 350 MPa.

FIG. 2 shows an isometric view of the annular beam 100, embodied in anon-pneumatic tire 150. The annular beam is connected to a central hub170 by a plurality of web spokes 180 that traverse the hub and beam inthe axial direction. The spokes have a sheet-like geometry in thisexample, yet other embodiments will be disclosed. The tire is loaded byand revolves around axis 140. A ground contacting portion 160 lies on aradially outward extent of the annular beam.

FIG. 3 shows an isometric view of an exemplary embodiment of anon-pneumatic tire 150, for which the outer diameter is around 10 to 13inches and the width in the axial direction is around 3 to 6 inches. Inthis figure, the entire structure has been designed such that it issuitable for one-shot high pressure injection molding. Thus, draftangles and radii for thermoplastic molding have been added.Additionally, the structure has been designed with wall thicknessesappropriate for this type of production process. Circumferentiallycontinuous bands 120 and 130, and web structure 110 have therefore beendefined accordingly. These features do not intend to limit the scope ofthis application in any way. They are provided to illustrate oneparticular best practice as applied to one specific production process.

FIG. 4 shows an isometric view of a second exemplary embodiment of atire 200 in accordance with the invention. The annular beam includesthree circumferentially continuous bands and a first web structure 240between band 210 and band 220, and a second web structure 250 betweenband 220 and band 230. This type of design is particularly suited forapplications having a larger outer diameter, along with a larger loadcapacity. As will be disclosed, such design freedoms enable the tiredesigner to tune the stiffness of the annular beam by geometry changes,as opposed to adding plies or other composite materials.

Many different annular beam web structures are possible. While notexhaustive, additional patterns are shown in FIGS. 5, 6, and 7 . In eachof these figures, a first circumferentially continuous band 120 lies ona radially inward extent of a web structure 110, with a secondcontinuous band 130 lying on a radially outward extent of the webstructure, and a tread contacting portion 160 lying on a radiallyoutward extent of the continuous band 130. FIG. 5 shows a cross sectionin the equatorial x-z plane of webbing consisting of column elements ofalternating inclination. FIG. 6 shows a web structure with a circulargeometry. FIG. 7 shows webbing having a geometric pattern of triangularand hexagonal elements. These web structures, and others that resultfrom combinations and additions of these designs, are intended to fallwithin the scope of this application, per the design approach andoptimization that will be disclosed.

Many different spoke web designs are possible. While not exhaustive,four general patterns are shown in the tire x-z equatorial plane inFIGS. 8, 9, 10 , and 11. In each figure, web spokes 180 connect a hub170 to an annular beam 100. FIG. 8 shows a radial pattern. FIG. 9 is adesign in which pairs of spokes are connected. FIG. 10 shows a diagonalpattern. FIG. 11 gives a curved radial design, in which the spokes havemodest departures from the radial direction. The spokes can also beinclined relative to the axial direction and otherwise designed tofacilitate demolding, which is most naturally done in the axial sense inthe spoke web region. These designs, and others that result fromcombinations of these, are all intended to fall within the scope of thisinvention.

A preferred design practice will now be disclosed. The inventor hasfound relationships that relate the annular beam shear stiffness tobending stiffness, such that excellent contact patch behavior isachieved. This is done within the confines of a monolithic annular beamcomposed of a homogeneous material, yet having the aforementioned webgeometry. When connected to a central wheel via webbed spokes, alsopreviously described, the structure has a high load to mass ratio, yetkeeps the simplicity of a homogeneous material, with no need forinextensible membranes or other composites or reinforcing elements.

A preferred tire contact pressure is substantially constant through thelength L of the contact patch. To achieve this, an annular beam ofradius R should be designed such that it develops a constant pressurewhen deformed to a flat surface. This is analogous to designing astraight beam which deforms to a circular arc of radius R when subjectedto a constant pressure which is equal to the contact pressure of theaforementioned annular beam. However, a homogeneous beam of solid crosssection does not behave like this. To create this desired performance,beam bending stiffness and beam shear stiffness can be intelligentlydesigned using a web geometry. A method for doing so will now bedisclosed, using standard nomenclature employed in typical engineeringtextbooks.¹

Equation 1 gives the relationship of shear force variation to an applieddistributed load on a differential beam element:

$\begin{matrix}{{- \frac{dV}{dx}} = W} & (1)\end{matrix}$Where:

V=transverse shear force

W=Constant distributed load per unit length

x=beam length coordinate

The deflection of such a beam due to shear deformation alone can beestimated by combining Equation 1 with other known relationships. Thisstep is not taken by those of ordinary skill in the art of mechanics, asshear deflection is usually ignored in calculating beam deflections.Adding relations between shear force, shear stress, shear modulus, andcross-sectional area, Equation 2 can be derived:

$\begin{matrix}{\frac{d^{2}z}{d^{2}x} = \frac{W}{GA}} & (2)\end{matrix}$Where:

G=beam shear modulus

A=effective beam cross sectional area

z=transverse beam deflection

For small deflections,

$\frac{d^{2}z}{d^{2}x}$is equal to the inverse of the beam deformed curvature. Making thissubstitution, which is not made by one of ordinary skill in the art, andconsidering a beam of unit depth, one obtains Equation 3:

$\begin{matrix}{P = \frac{GA}{R}} & (3)\end{matrix}$Where:

G=beam shear modulus

R=deformed beam radius of curvature

A=effective beam cross sectional area, with unit depth

P=Constant distributed pressure, with unit depth

Equation 3 is very important. A straight beam of shear modulus G andeffective cross sectional area A, subjected to homogeneous pressure P,will deform into the shape of an arc of radius R, provided sheardeflection predominates.

Similarly, an annular beam of radius R, designed such that sheardeformation predominates, that is deflected against a flat contactsurface, will develop a homogeneous contact pressure P. This is shown inFIG. 12 , in which an annular beam according to the invention is placedin contact between two parallel plates. Along a design contact length L,a constant contact pressure is developed. As mentioned, this isanalogous to the straight beam that deforms to the shape of an arc ofradius R, which subjected to the homogeneous pressure P.

A constant pressure through design contact length L is a highly desiredperformance attribute. An annular beam with this attribute could beadvantageously employed in conveyor belt systems, and wheels used intrack drive systems which have opposing contact patches. It isparticularly useful when embodied in a non-pneumatic tire, as shown inFIG. 13 . Here, the annular beam is attached to a central hub by a spokeweb geometry. When a design load is applied at the hub, the beam deformsover a design contact patch length L, develops a homogeneous contactpressure, and passes the load to the hub via tension in the web spokes.

The present invention achieves the desired performance of FIG. 13 bydesigning the annular beam with a web structure that generally traversesthe beam in the axial direction. With this design feature, sheardeflection can be larger than bending deflection, and the desiredcontact patch performance can be obtained.

The inventor has found that analysis of a straight beam is lesscumbersome than an annular beam; therefore the first part of the designprocess employs a straight beam geometry subjected to a constantpressure, in order to design the web structure. Final designverification then includes a complete model, as will be disclosed.

Towards this end, the first step in developing a design process is tocalculate the deflection due to bending and the deflection due to shearof a simply supported straight beam subjected to a constant pressure.Equation 4 gives the center deflection due to bending; Equation 5 givesthe center deflection due to shear; Equation 6 solves for sheardeflection divided by bending deflection:

$\begin{matrix}{z_{b} = {\frac{5}{384}\frac{{PL}^{4}}{EI}}} & (4)\end{matrix}$ $\begin{matrix}{Z_{S} = {\frac{1}{4}\frac{{PL}^{2}}{GA}}} & (5)\end{matrix}$ $\begin{matrix}{\frac{z_{s}}{z_{b}} = {19.2\frac{EI}{L^{2}}\frac{1}{GA}}} & (6)\end{matrix}$Where:

z_(b)=beam center deflection due to bending

z_(s)=beam center deflection due to shear

L=beam length, which is about equal to the tire contact length

E=beam tensile modulus

I=beam moment of inertia

The result of Equation (6) is a dimensionless term that describes animportant aspect of the annular beam behavior. It is a geometrical termthat, for homogeneous materials, is independent of modulus. Asz_(s)/z_(b) becomes larger, shear deflection predominates. As sheardeflection predominates, Equation (3) becomes valid and the desiredperformance of FIG. 13 is obtained.

Shear deflection is usually assumed to be large compared to bendingdeflection, and shear deflection is neglected. Consequently, one ofordinary skill in the art of mechanics is not familiar with the resultof Equation (6). Beam bending stiffness must be relatively high, andbeam shear stiffness must be relatively low in order to have z_(s)/z_(b)be acceptably high.

The next step is to define a procedure to relate beam design variablesto the terms of Equation 6. This is accomplished by a Finite ElementAnalysis (FEA) on the geometry of the annular beam. While many standardFEA packages are suitable for this analysis, Abaqus was used foranalyses in this application. To analyze the annular beam design, thefollowing steps were followed:

-   -   Representation of the annular beam as a 2D straight beam.    -   Definition of the beam geometry. Beam length, L, equals the        design tire contact patch length in the x direction. The beam        webbing design and associated dimensions are also included as        model parameters.    -   Two small-deflection plane stress 2D FEA calculations are        performed.        -   i. Effective shear modulus calculation. This is done by            constraining the bottom side of the beam and applying a            uniform shear force to the top face.        -   ii. Total center beam deflection calculation. This is done            by simply supporting the beam and applying a uniform            pressure to the top face such that the beam deflects in the            z axis.

FEM used in (i) and (ii) are illustrated in FIGS. 14 and 15 ,respectively.

From the FEM output of the model of FIG. 15 , the beam effective shearmodulus can be determined:

$\begin{matrix}{G_{eff} = \frac{\tau}{\gamma}} & (7)\end{matrix}$Where:

τ=applied shear stress at beam top surface

γ=average angular deformation across beam webbing in the z direction.

G_(eff)=effective beam shear modulus

The effective shear modulus calculation is used with Equation (5) tocalculate z_(s), the beam center deflection due to shear. For a unitdepth assumption with the plane stress FEM, the effective beam crosssectional area A for shear deformation calculation equals the beamwebbing thickness in the z direction. The webbing between the twocontinuous bands is much softer in shear than the bands; therefore, theshear strain is higher in the webbing. Since beam deflections in sheardepend on the largest shear angle which lies in the beam section, theeffective beam cross section area is calculated relative to the totalwebbing thickness in the z axis.

The second FEA model calculates the total center beam center deflection.By subtracting the deflection due to shear, the deflection due tobending is found. Equation (4) is rearranged to calculate the effectivemoment of inertia of the beam:

$\begin{matrix}{z_{b} = {z_{t} - z_{s}}} & (8)\end{matrix}$ $\begin{matrix}{I_{eff} = {\frac{5}{384}\frac{{PL}^{4}}{{Ez}_{b}}}} & (9)\end{matrix}$Where:

z_(t)=total beam center deflection from FEA calculation

I_(eff)=effective beam moment of inertia

For homogeneous, isotropic materials, the shear modulus and tensilemodulus are related by Poisson's ratio, as given in Equation (10):

$\begin{matrix}{G = \frac{E}{2\left( {1 + \upsilon} \right)}} & (10)\end{matrix}$Where:

υ=Poisson's ratio

E=tensile modulus

G=shear modulus

With the effective beam moment of inertia and the effective beamcross-sectional area known, the designer can then model the performanceof an annular beam of the same design when used in the context of thepresent invention. The design procedure is then used to optimize thebeam webbing design by employing Equation (6) to calculate the ratio ofshear deflection to bending deflection for any design contact patchlength L.

An efficient way to employ the FE models of FIGS. 14 and 15 is tonormalize the tensile modulus to 1.0. The shear modulus is then definedby Equation (10). In most FE codes, this value can be input as modulus,along with a value for Poisson's ratio. For many thermoplastic and castelastomers, this dimensionless value is approximately 0.45. Thus, for anormalized tensile modulus, the shear modulus equals about 0.344. Astraight-forward example of this design method is now provided.

A simple web design is used in order to illustrate the design procedure.Other web designs could consist of additional variables and more complexgeometries. The FEA beam model geometry of FIG. 16 depicts a webstructure 110 consisting of circular cut-outs of diameter Dc, with apace in the x direction between each cut-out of distance p. This webstructure is between continuous bands 120 and 130 of thickness t in thez direction. As the distance between the web cutouts decreases, theamount of material removed increases. Therefore, the effective shearmodulus decreases. As the distance p approaches cut-out diameter Dc, theshear stiffness approaches zero.

The bending stiffness also decreases as webbing material is removed.However, the bending stiffness does not decrease as rapidly as does theshear stiffness because of the maintenance of the two solid bands ofthickness t at the beam outer fibers. A large moment of inertia,therefore, is maintained. For this reason, the shear deflection willbecome large compared to the bending deflection as the webbing pace isdecreased.

FIG. 17 shows FEA results for effective shear modulus G_(eff), and theratio of shear deflection to bending deflection z_(s)/z_(b). asfunctions of the webbing pace p. The FE models used a tensile modulusnormalized to 1.0, and therefore a shear modulus of 0.344. For thisexample, Dc=10 mm, t=2 mm, and L=60 mm.

For a webbing pace that is very large, the effective shear modulus willasymptotically approach the isotropic value of 0.344. As the webbingpace decreases to 10.0, the effective shear modulus approaches 0. Thereason for this is that, when the webbing pace equals the radius of thecutout, the beam becomes discontinuous through the thickness. Theresulting structure would be free to displace in shear with no shearforce, and the effective shear modulus would be zero. With a webbingpace of 12 mm, there is 2 mm of material between the webbing cutouts,and the resulting G_(eff) is 0.033, which represents a decrease of morethan an order of magnitude compared to the isotropic shear modulus of0.344.

For a webbing pace above 20 mm, the bending deflection becomes largerthan the shear deflection. For a webbing pace of 14 mm or below, theshear deflection becomes significantly larger than the bendingdeflection. In this design region, where shear deflection is highcompared to bending deflection, the positive performance attributesshown in FIG. 12 will be attained.

For a specific example, a small tire having an outer radius of 125 mmcan be designed using an annular beam as in FIG. 16 , with p=12 mm, t=2mm, and Dc=10 mm. A design contact patch pressure of 0.20 MPa (30 psi)is used as a design input, along with a contact patch length L=60 mm. Toattain this value in the design, Equation (3) is used to solve for theneeded effective shear stiffness. With a unit depth, the effective sheararea of the annular beam used in the FEM results from FIG. 17 is thewebbing thickness in the z direction times unit depth. Thus:

$\begin{matrix}{{P = {\frac{0.2{N\left( {1{mm}{unit}{depth}} \right)}}{{mm}^{2}} = \frac{{{Geff}\left( {10{mm}} \right)}\left( {1{mm}{unit}{depth}} \right)}{125{mm}}}}{{Geff} = {2.5{MPa}}}} & (11)\end{matrix}$

For a webbing pace=12 mm, G_(eff)=0.033 for the normalized modulus ofFIG. 17 . Therefore, the desired actual shear modulus is:

$\begin{matrix}{G = {\frac{2.5{MPa}}{0.033} = {75{MPa}}}} & (12)\end{matrix}$

From Equation (10) the tensile modulus of the material is should beabout 210 MPa.

Therefore, by choosing a homogeneous material with a tensile modulus of210 MPa, a contact patch pressure of 0.20 MPa will be obtained. Thecontact length of 60 mm has already been defined as an input to the FEanalyses. The design load carrying capacity is defined by multiplyingpressure by contact area. For a lawn mower tire, a design load of 1115 N(250 lbs) is required. Thus:

$\begin{matrix}{W = {\frac{F}{LP} = {\frac{1115N}{\left( {60{mm}} \right)\left( {0.2{MPa}} \right)} = {93{mm}}}}} & (13)\end{matrix}$Where:

F=design load

L=tire contact patch length in X direction=beam length in FEcalculations

P=desired contact pressure

W=contact patch width

The second phase involves verification of the annular beam design with a2D plane stress FEM of a non-pneumatic tire. FIG. 17 shows the deformedgeometry of a non-pneumatic tire according to the invention that has aweb spacing of 12 mm, with the annular beam geometry as approximatelydefined in FIG. 16 , and material properties about equal to thosecalculated above.

At a load of 1115 N (250 lbs), and a contact width in the depthdirection of 95 mm, the predicted contact force is equally distributedthrough the contact length, as shown in FIG. 18 . This result shows thata web spacing of 12 mm, and hence a z_(s)/z_(b) value of 3.8, doesindeed give a shear deflection that is sufficiently high compared tobending deflection. Equation 3 becomes operative, and the contact patchpressure is essentially homogeneous.

Additional work by the inventor has shown that a z_(s)/z_(b) value of atleast about 1.2, and preferably above 2.0, is necessary to obtain arelatively homogeneous pressure throughout the contact length.

This approximate tire size of 10 inch diameter×4 inch width (100 mm) isa common size used for lawn mower and other caster tires. A design loadof 250 lbs per tire is very acceptable in this market.

FIG. 18 also shows the Von Mises stress distribution in the deformedgeometry. The graduated gray scale of 0 MPa to 8 MPa shows that maximumvalues of less than 7.4 MPa are obtained with this web geometry at thedesign load. In order to sustain cyclic stress of this magnitude, a goodflex fatigue performance is required. Further, tires are often subjectedto impact loads and road hazard impacts that result in very high localstrains. A material having a high level of tensile strain at break isdesired. For some applications, a flex fatigue limit of 4 MPa issufficient, with a value above 7 MPa preferred. A tensile strain atbreak of above 100% is also preferred. These must be combined with atensile modulus of about 200 MPa.

Examples of materials having these physical characteristics includethermoplastic urethanes such as Elastollan S98A and thermoplasticco-polymers such as Hytrel G5544 and 5556. Some versions of unfilled,plasticized, toughened engineering plastics attain similar capabilities,such as Zytel 351 PHS. Examples of cast polyurethane thermoset materialsattaining these characteristics are COIM PET-95A and PET-60D, when curedwith curative MBOCA or MCDEA. These are just a few of many examples andare not meant to limit the scope of invention in any way.

The design procedure covered in the preceding paragraphs is suitable forthe design and optimization of a large variety of geometries fallingwithin the scope of this application. The above example was a small tiresuitable for a lawnmower, with a relatively short contact length.However, larger tires will have longer contact lengths. Equation (6) hasthe contact length as a squared term in the denominator. To maintain anacceptably high z_(s)/z_(b), one strategy is to increase the beam momentof inertia, which is in the numerator of Equation (6).

Increasing the moment of inertia involves increasing the distancebetween the circumferentially continuous bands and/or increasing theband thickness. FIG. 4 showed one way to do this, while maintaining therelatively constant wall thicknesses required by thermoplasticinjection. The tire shown in FIG. 4 has a total annular beam radialthickness that has been greatly increased, while wall thicknesses havebeen only moderately increased. The higher design load of this tire hasbeen achieved largely due to changes in the geometry of the annularbeam, with no need for inextensible membranes, plies, or composites.

Additional work by the inventor has shown that certain elastomericmaterials exhibit favorable non-linear stress vs. straincharacteristics. One preferred embodiment involves the choice of amaterial having a very non-linear material behavior, for which thesecant modulus decreases with increasing strain. From the definitionearlier provided, “modulus” is the initial slope of the stress vs.strain curve, often termed “Young's modulus” or “tensile modulus.”Preferred materials have a high Young's modulus that is much greaterthan the secant modulus at 100% strain, which is often termed “the 100%modulus.” This nonlinear behavior provides efficient load carryingduring normal operation, yet enables impact loading and large localdeflections without generating high stresses.

Some thermoset and thermoplastic polyurethanes have this materialbehavior. An example of such a favorable material is shown in FIG. 20 .The measured stress vs. strain curve of COIM's PET-95A, with curativeMCDEA, has a Young's modulus of 205 MPa. However, the secant modulus at100% strain is only 19 MPa. This is a favorable attribute for thepresent invention; when following the design principles earlierdisclosed, the tire normally operates in the 2 to 5% strain region. Inthis region, the material is moderately stiff and the slope of thestress vs. strain curve is fairly constant. However, if localdeformation occurs due to road hazards or impacts, the material iscapable of large strains, without generation of high stresses. Thisminimizes vehicle shock loading, and enhances tire durability.

Those skilled in the art of elastomer chemistry do not recognize thepotential of this material behavior. Elastomers are often used in areasof high imposed strains. As such, testing protocol typically focuses onthe performance at high strains, such as 100%, 200%, or more. Mechanicaldesigns that carry load in tension and bending typically do not use onehomogeneous elastomer—they employ reinforcements as well. This inventionopens this new design space by leveraging this material non-linearitywith a favorable mechanical design.

FIG. 21 shows an exemplary embodiment of a tire according to theinvention for use in the construction industry. This large tire wasdesigned using the above principles, along with the non-linear materialbehavior of COIM's PET-95A. With an outer diameter of 1500 mm (60 in)and width of 530 mm (21 in), the tire is capable of a design load of10,000 kg (22,000 lbs.). It is a replacement for a pneumatic tire/wheelin the dimension 20.5×25. In this embodiment, the tread, circumferentialband, spokes, and hub are all composed of PET-95A. The contact pressurewas designed at 0.5 MPa (74 psi). The circumferentially continuous bandshave a thickness of 15 mm (0.6 inches), and the web thickness is 85 mm(3.4 inches). Simulated impact loadings do not result in high materialstresses, due to the non-linear stress vs. strain curve, while materialstrains are in the 2-5% range during normal operation. Total weight ofthis non-pneumatic tire/wheel is 275 kg (595 lbs.). This weight is lessthan the mounted assembly weight of pneumatic tires of similar size anddesign load. Vertical stiffness and contact pressure are designed to besimilar to the comparable pneumatic tire.

A tread contacting region can be affixed to a radially outer extent ofthe annular beam in many ways. The tread can be composed of a differentmaterial and can extend around the part or all of the axial extents ofthe annular beam. This is shown in FIG. 22 . The tread is designed suchthat the annular beam web portion 110 is encapsulated and covered by thetread material 190. Additionally, the lateral extents of the web areahave been modified to mate with and indeed provide mechanicalinterlocking for the tread material. In this design, the web portion isprotected from rocks, dirt, or other debris, This illustrates onepreferred embodiment, yet is not intended to limit the scope of theclaim of this application.

The design procedure previously described can be represented as adecision flow-chart, shown in FIG. 23 . Using this approach, a humandesigner or a computer algorithm can optimize a given annular beam webdesign for a specific design requirement as embodied in a non-pneumatictire.

While certain exemplary embodiments have been described and shown in theaccompanying drawings, it is to be understood that such embodiments aremerely illustrative of and not restrictive on the broad invention, andthat this invention not be limited to the specific constructions andarrangements shown and described, since various other changes,combinations, omissions, modifications and substitutions, in addition tothose set forth in the above paragraphs, are possible. Those skilled inthe art will appreciate that various adaptations, combinations, andmodifications of the just described embodiments can be configuredwithout departing from the scope and spirit of the invention.

REFERENCES

-   1. Muvdi, B. B., McNabb, J. W., (1980). Engineering Mechanics of    Materials, Macmillan Publishing Co., Inc., New York, N.Y., “Shear    and Bending Moment in Beams,” pp 23-31, and “Deflections of Beams”,    pp 266-333.

What is claimed is:
 1. A non-pneumatic tire comprising: a monolithicannular beam; a hub spaced radially from the annular beam; and aplurality of spokes that extend from the annular beam to the hub suchthat, when the non-pneumatic tire is loaded, upper ones of the spokesabove an axis of rotation of the non-pneumatic tire are in tension; theannular beam being elastomeric, the annular beam being configured todeflect more by shearing than by bending at the contact patch of thenon-pneumatic tire; and wherein the non-pneumatic tire comprisesthermoplastic elastomeric material forming at least part of the spokesand having a flex fatigue limit of at least 4 MPa according to ASTMD671.
 2. The non-pneumatic tire of claim 1, wherein each of the spokesextends from the annular beam to the hub without being intersected atmore than one point.
 3. The non-pneumatic tire of claim 2, wherein eachof the spokes extends from the annular beam to the hub without beingintersected.
 4. The non-pneumatic tire of claim 1, wherein each of thespokes extends from the annular beam to the hub without being connectedto the other ones of the spokes at more than one point between theannular beam and the hub.
 5. The non-pneumatic tire of claim 1, whereineach of the spokes is unconnected to the other ones of the spokes fromthe annular beam to the hub.
 6. The non-pneumatic tire of claim 3,wherein the flex fatigue limit of the thermoplastic elastomeric materialis at least 7 MPa.
 7. The non-pneumatic tire of claim 3, wherein aYoung's modulus of the thermoplastic material is at least 90 MPa.
 8. Thenon-pneumatic tire of claim 3, wherein the thermoplastic elastomericmaterial is a thermoplastic co-polymer.
 9. The non-pneumatic tire ofclaim 1, wherein the thermoplastic elastomeric material forms at leastpart of the annular beam.
 10. The non-pneumatic tire of claim 9, whereinthe annular beam and the spokes are entirely formed of the thermoplasticelastomeric material, thermoplastic elastomeric material.
 11. Thenon-pneumatic tire of claim 1, wherein the annular beam comprises: anouter annular portion; an inner annular portion; and a shearing annularportion between the outer annular portion and the inner annular portionof the annular beam.
 12. The non-pneumatic tire of claim 11 wherein theannular beam comprises a plurality of voids distributed in acircumferential direction of the annular beam.
 13. A process formanufacturing a non-pneumatic tire that comprises: a monolithic annularbeam; a hub spaced radially from the annular beam; and a plurality ofspokes that extend from the annular beam to the hub such that, when thenon-pneumatic tire is loaded, upper ones of the spokes above an axis ofrotation of the non-pneumatic tire are in tension; the annular beambeing elastomeric, the annular beam being configured to deflect more byshearing than by bending at the contact patch of the non-pneumatic tire;the process comprising: forming the annular beam; and forming thespokes; and wherein the non-pneumatic tire comprises thermoplasticelastomeric material forming at least part of the spokes.
 14. Theprocess of claim 13, wherein each of the spokes is unconnected to theother ones of the spokes from the annular beam to the hub.
 15. Theprocess of claim 14, wherein the thermoplastic elastomer material has aflex fatigue limit of at least 4 MPA according to ASTM D671.
 16. Anon-pneumatic tire comprising: an annular beam configured to deflect ata contact patch of the non-pneumatic tire with a ground surface, theannular beam comprising: an outer annular portion; an inner annularportion: and a shearing annular portion between the outer annularportion and the inner annular portion of the annular beam; the outerannular portion, the inner annular portion, and the shearing annularportion of the annular beam being elastomeric, the annular beam beingconfigured to deflect more by shearing than by bending at the contactpatch of the non-pneumatic tire; a hub spaced radially from the annularbeam; and a plurality of spokes that extend from the annular beam to thehub such that, when the non-pneumatic tire is loaded, upper ones of thespokes above an axis of rotation of the non-pneumatic tire are intension; the annular beam being elastomeric, the annular beam beingconfigured to deflect more by shearing than by bending at the contactpatch of the non-pneumatic tire; and wherein the non-pneumatic tirecomprises thermoplastic elastomeric material forming at least part ofthe spokes and having a flex fatigue limit of at least 4 MPa accordingto ASTM D671.